Abstract
The non-commutative counterpart of the well-known Łukasiewicz propositional logic is developed, in strong connection with the algebraic theory of psMV-algebras. An extension by a new unary logical connective is also considered and a stronger completeness result is proved for this system.
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References
Baaz, M.: Infinite-valued Gödel logics with 0-1-projections and relativizations. In: GÖDEL'96-Logical Foundations of Mathematics, Computer Science and Physics; Lectures Notes in Logic 6, P. Hájek, Ed., Springer Verlag, 23–33 (1996)
Ceterchi, R.: On Algebras with Implications, Categorically Equivalent to Pseudo-MV Algebras. In: The Proceedings of the Fourth International Symposium on Economic Informatics Bucharest, Romania, 912–916 (1999)
Ceterchi, R.: Pseudo-Wajsberg Algebras. Multiple-Valued Logic 6, 67–88 (2001)
Chang, C.C.: Algebraic analysis of many valued logics. Trans. A.M.S. 88, 467–490 (1958)
Chang, C.C.: A new proof of the completeness of the Lukasiewicz axioms. Trans. A.M.S. 93, 74–80 (1959)
Cignoli, R., D'Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of many-valued Reasoning. Kluwer Academic Publ. Dordrecht (2000).
Dvurečenskij, A.: Pseudo MV-algebras are intervals in ℓ-groups. Journal of Australian Mathematical Society 72, 427–445 (2002)
Dvurečenskij, A.: States on Pseudo MV-algebras. Studia Logica 68, 301–327 (2001)
Font, J.M., Rodríguez, A.J., Torrens, A.: Wajsberg algebras. Stochastica 8, 5–31 (1984)
Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras: a noncommutative extension of MV-algebras. The Proceedings of the Fourth International Symposium on Economic Informatics, Bucharest, Romania 961–968, (1999)
Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras. Multi. Val. Logic 6, 95–135 (2001)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Acad. Publ. Dordrecht (1999)
Hájek, P.: On very true. Fuzzy Sets and Systems 124, 329–333 (2001)
Hájek, P.: Fuzzy logics with non-commutative conjunction. Journal of Logic and Computation, To appear
Hájek, P.: Observations on non-commutative fuzzy logic. Soft Computing, To appear
Hájek, P.: Fleas and fuzzy logic, draft.
Kühr, J.: Pseudo BL-algebras and DRℓ-monoids. Czechoslovak Math. Journal, To appear
Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Cl. III, 23, 30–50 (1930)
Rachunek, J.: A non-commutative generalization of MV-algebras. Czechoslovak Math. Journal 52, 255–273 (2002)
Rasiowa, H.: An algebraic approach to non-classical logic. North Holland Publ., Amsterdam, Polish Scientific Publ., Warszawa 1974
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Leuştean, I. Non-commutative Łukasiewicz propositional logic. Arch. Math. Logic 45, 191–213 (2006). https://doi.org/10.1007/s00153-005-0297-8
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DOI: https://doi.org/10.1007/s00153-005-0297-8